{% extends 'homepage.html' %}
{% block content %}

<p>The database consists of fields from three sources:
<ol>
<li>The PARI database from the Bordeaux PARI group
<li>Additional totally real fields of degrees from 6 to 10 computed by
  John Voight.
<li>Additional fields from John Jones-David Roberts database.
</ol>
</p>

<h3>Details of the fields contained in the database</h3>
<p>
<ol>
<li>PARI database discriminant ranges.  It is possible that the
  discriminant range depends further on the signature, but this is not
  shown here.
<p>
<table>
{% set row_class = cycler("odd", "even") %}
<tr>
<th>degree</th>
<th>minimum discriminant</th>
<th>maximum discriminant</th>
</tr>
<tr class="{{ row_class.next() }}"><td>1<td>\(1\)<td>\(1\)</tr>
<tr class="{{ row_class.next() }}"><td>2<td>\(-10^6\)<td>\(10^6\)</tr>
<tr class="{{ row_class.next() }}"><td>3<td>\(-10^6\)<td>\(2\cdot10^6\)</tr>
<tr class="{{ row_class.next() }}"><td>4<td>\(-10^6\)<td>\(10^6\)</tr>
<tr class="{{ row_class.next() }}"><td>5<td>\(-10^6\)<td>\(2\cdot10^7\)</tr>
<tr class="{{ row_class.next() }}"><td>6<td>\(-10^6\)<td>\(10^7\)</tr>
<tr class="{{ row_class.next() }}"><td>7<td>\(-11841551\)<td>\(149324209\)</tr>
</table>
</p>
<li>Voight database additional discriminant ranges.  For degrees up to
  9, the database contains all totally real fields with discriminant
  in the given range.
<p>
<table>
{% set row_class = cycler("odd", "even") %}
<tr>
<th>degree</th>
<th>minimum discriminant</th>
<th>maximum discriminant</th>
</tr>
<tr class="{{ row_class.next() }}"><td>6<td>\(1\)<td>\(16771805\)</tr>
<tr class="{{ row_class.next() }}"><td>7<td>\(1\)<td>\(213873729\)</tr>
<tr class="{{ row_class.next() }}"><td>8<td>\(1\)<td>\(2556640000\)</tr>
<tr class="{{ row_class.next() }}"><td>9<td>\(1\)<td>\(25405254289\)</tr>
<tr class="{{ row_class.next() }}"><td>10<td>\(1\)<td>\(7623696325213\)</tr>
</table>
</p>
<li>Jones-Roberts database additional fields.
<p>
The degree of a field is given by \(n\).
<table>
{% set row_class = cycler("odd", "even") %}
<tr class="{{ row_class.next() }}"><td>Fields unramified outside \(\{2,3\}\)
 with \(n\leq 7\)
</tr>
<tr class="{{ row_class.next() }}"><td>Fields ramified at only one prime \(p\) with \(p<200\) with \(n\leq 9\) </tr>
<tr class="{{ row_class.next() }}"><td>Fields ramified at only two primes \(p\lt q \leq 5\) with \(n\leq 8\) </tr>
</table>
</p>
<p>
The remaining cases, the bound depends on the Galois group.  Galois groups
are given in the form \(n\)T\(t\) where \(n\) is the degree and \(t\)
it the T-number.  The bound \(B\) is for the root discriminant, so a
bound of \(B\) means the discriminant \(D\) satisfies \(|D|\leq B^n\).
</p>
<p>
<table>
{% set row_class = cycler("odd", "even") %}
<tr>
<th>\(n\)T\(t\)</th>
<th>\(B\)</th>
</tr>
<tr class="{{ row_class.next() }}"><td>7T3<td>\(26\)</tr>
<tr class="{{ row_class.next() }}"><td>7T5<td>\(38\)</tr>
<tr class="{{ row_class.next() }}"><td>8T3<td>\(20\)</tr>
<tr class="{{ row_class.next() }}"><td>8T5<td>\(50\)</tr>
<tr class="{{ row_class.next() }}"><td>8T15<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T18<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T22<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T26<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T29<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T32<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T34<td>\(15\)</tr>
<tr class="{{ row_class.next() }}"><td>8T36<td>\(15\)</tr>
</table>
</ol>
</p>

{% endblock %}
</html>
